NIST Technical Note 2034 Defning ‘kind of quantity’ David Flater This publication is available free of charge from: https://doi.org/10.6028/NIST.TN.2034 NIST Technical Note 2034 Defning ‘kind of quantity’ David Flater Software and Systems Division Information Technology Laboratory This publication is available free of charge from: https://doi.org/10.6028/NIST.TN.2034 February 2019 U.S. Department of Commerce Wilbur L. Ross, Jr., Secretary National Institute of Standards and Technology Walter Copan, NIST Director and Undersecretary of Commerce for Standards and Technology Certain commercial entities, equipment, or materials may be identifed in this document in order to describe an experimental procedure or concept adequately. Such identifcation is not intended to imply recommendation or endorsement by the National Institute of Standards and Technology, nor is it intended to imply that the entities, materials, or equipment are necessarily the best available for the purpose. National Institute of Standards and Technology Technical Note 2034 Natl. Inst. Stand. Technol. Tech. Note 2034, 7 pages (February 2019) CODEN: NTNOEF This publication is available free of charge from: https://doi.org/10.6028/NIST.TN.2034 NIST Technical Note 2034 1 Defning ‘kind of quantity’ David Flater 2019-02-06 This publication is available free of charge from: https://doi.org/10.6028/NIST.TN.2034 Abstract The defnition of ‘kind of quantity’ given in the International Vocabulary of Metrology (VIM), 3rd edition, does not cover the historical meaning of the term as it is most commonly used in metrology. Most of its historical meaning has been merged into ‘quantity,’ which is polysemic across two layers of abstraction. I propose a model of ‘kind of quantity’ that is compatible with its historical meaning and distinguishable from ‘quantity in a general sense.’ Preface In this report I follow the convention that was provided in [1]: Terms referring to terms themselves are delimited by double quotes (e.g. “measurement” is an 11-letter word); terms referring to concepts are delimited by single quotes (e.g. ‘measurement’ is a key concept of metrology); fnally, terms with their usual referents are not delimited (e.g. measurement is a key process of metrology). Double quotes are also used for verbatim quotations included inline. Bold and italic fonts are used or retained where necessary for consistency with the text of the International Vocabulary of Metrology (VIM) [2] and other cited references. 1 Introduction The concept ‘kind of quantity’ has historically served to explain when and how it makes sense to compare two quantities, put them in order of magnitude, or add them together. Broadly speaking, these operations are meaningful when the quantities are of the same kind, otherwise not. As important as this concept is, many scientists remain uncomfortable with its defnition, particularly how to distinguish it from other ways of categorizing quantities. In this report I propose a model of ‘kind of quantity’ that makes it compatible with its historical meaning and distinguishable from closely related concepts such as ‘quantity in a general sense.’ The remaining sections are organized as follows. Section 2 reviews the evolution of defnitions of ‘kind of quantity’ in standards and related work. Section 3 explains the meanings of elements of the model. Section 4 provides the model itself. Section 5 continues with additional discussion of the model. Finally, Section 6 concludes. NIST Technical Note 2034 2 2 History of “kind of quantity” and related terms Maxwell referred to kinds of quantities in his preliminary text on the measurement of quantities, writing “There must be as many di erent units as there are di erent kinds of quantities to be measured” [3]. An ontology by Ren´e Dybkaer provides a chronological list of relevant defnitions appearing from 1951 through 2007 [4, Table 6.5]. In it, we can see an apparent schism between ‘kind of quantity’ seen as an abstraction, generalization, category, collection, or class of quantities and ‘kind of quantity’ seen instead as an element of information or aspect that quantities would have. A representative example of the former, as translated from DIN 1313 [5] by Dybkaer, is: “(Gr¨oßenart) collection of quantities which are considered to This publication is available free of charge from: https://doi.org/10.6028/NIST.TN.2034 be qualitatively alike and for which it is meaningful to add quantity values, independent from a quantity system to which they may belong.” A representative example of the latter is the VIM 3 defnition: “aspect common to mutually comparable quantities.” ‘Kind of quantity’ was the primary organizing concept of the reference Quantities and Units in Clinical Chemistry (Recommendations 1966 et seq.) [6, 7]. Its descendant, the “silver book” [8], is organized by quantity dimensions frst, then by kinds of quantities. Thus it is evident that the term has remained in continuous use in clinical chemistry from 1966 to the present day. The frst release of the VIM [9] distinguished ‘quantities in a general sense’ from ‘specifc quantities’ and stated that “Quantities which are mutually comparable may be grouped together into categories of quantities.” The 1987 corrections to VIM 1 added text to say that quantities that are mutually comparable are “also called quantities of the same kind.” The second edition of the VIM [10] replaced “specifc quantity” with “particular quantity” (Def. 1.1 Note 1 and Def. 1.7) and replaced “mutually comparable” with “can be placed in order of magnitude relative to one another” (Def. 1.1 Note 2). The third edition of the VIM [2] defned ‘kind of quantity’ as “aspect common to mutually comparable quantities,” and it replaced the entire discussion of general versus particular quantities and categories of quantities with “The generic concept ‘quantity’ can be divided into several levels of specifc concepts, as shown in the following table. .” The examples in the lowest level are referred to in passing as “individual quantities.” Mari [11] performed an analysis of the VIM’s ‘kind of quantity’ and related concepts using an object-oriented viewpoint. Dybkaer [12] rejected Mari’s analysis, emphasizing that the VIM must be understood in the context of International Organization for Standardization (ISO) 704 [13] and ISO 1087-1 [14], standards for terminology work that are referenced by the VIM. [12] proceeded to explain that ‘kind of quantity’ occupies the role that [13, 14] refer to as “criterion of subdivision.” [1] used the terms “general quantity” and “individual quantity” where VIM 2 used “quantity in a general sense” and “particular quantity.” [12] also used “individual quantities.” In the following sections I will use “general quantity” and “individual quantity” consistently with [1]. Its further subdivision of ‘individual quantities’ into ‘quantities of objects’ (which are called “addressed quantities” in [7]) and ‘values of quantities’ is not needed for this discussion. 3 Model preliminaries It is not my intention to advocate for either concept-oriented or object-oriented viewpoints or language. What is important is that the necessary vocabulary for conceptual modelling is established, and for most purposes either type of language would do. Table 1 provides a cross-reference for the approximately equivalent notions from the concept-oriented ISO 704 + 1087-1, the object-oriented Unifed Modeling Language (UML) [17]1 , and generic set theory. 1Although ISO/IEC 19501:2005 (UML version 1.4.2) might be superseded by ISO/IEC 19505-1:2012 and 19505-2:2012 (UML version 2.4.1), the older version contains a simple glossary of the terms that are relevant here [17, p. 411] while the newer version does not. NIST Technical Note 2034 3 Table 1: Approximate equivalences between concept-oriented, object-oriented, and set-theoretic language. ISO 704 + 1087-1 UML Generic set theory Notes Object Object Object “Objects are perceived or conceived; are abstracted or conceptualized into concepts” [13]. Other names for ob- jects include individuals, atoms, and ur-elements [15, 16]. Concept Class Set “Concepts depict or correspond to objects or sets of ob- jects” [13] (see notes) Instance Member / element “An object is an instance of a class” [17]; “single object in an extension ” [12]. “Instance” is used in passing in [13, §6.4.3] and was adopted in [12] for use with ISO 704 This publication is available free of charge from: https://doi.org/10.6028/NIST.TN.2034 + 1087-1. Generic concept Superclass Superset “The superordinate concept in a generic relation” [13]; “The generalization of another class” [17] Specifc concept Subclass Subset “The subordinate concept in a generic relation” [13]; “The specialization of another class” [17] (no term) Metaclass Set (of sets) “A class whose instances are classes” [17] When an object-oriented modelling language is used as a representation for conceptual modelling, a class represents a concept.2 Coincidentally, [13, §5.1] uses the word “class” while explaining the nature of concepts: “Through observation and a process of abstraction called conceptualization, objects are categorized into classes, which correspond to units of knowledge called concepts, which are represented in various forms of communication (object ! concept ! communication).” Regardless of the language used, the formal structure described below is maintained. However, to simplify the text, I will provide only the terms from the UML column of Table 1; for example, writing only “class” where one might instead write “concept/class/set.” A class has both intent and extent (or intension and extension). The intent consists of necessary and suÿcient conditions to deduce that an object is an instance of the class. The extent consists of those objects that are instances. For example, the intent of the class ‘bat’ is provided by the defnition of this subclass of mammals, while the extent is all of the bats in the world. The intent of a subclass consists of the intent of its superclass (which it “inherits”) plus at least one additional, narrowing condition.